Determinant of a metric in the three dimensional positive?

Question

In the three dimensional Euclidean space, given any basis $\{e_1, e_2, e_3\}$. Then, does the determinant of the metric $g_{ij}=e_i \cdot e_j$ always have positive value? It was easy to show in the 2 dimension, but I cannot find a way to prove it in the three dimension.

Answer 1

Let $A$ denote the matrix whose columns are $e_1,e_2,e_3$. We then note that $G = A^TA$. It follows that $$ \det(G) = \det(A^T) \det(A) = \det(A)^2 $$ so yes: if $\{e_1,e_2,e_3\}$ is a basis, then $\det(G)$ is positive.